УрМатФиз / УрМатФиз с теорией

!0 # r:

a

nr

kr

a2

2

J

a ! J

a

! rdr = nk 2 J0

n :

' ( ) 2.5.49

J0

kr

! r, -

a

r !0 # )

a

nr

a2

Z f r z J0

2

a ! rdr = fn z 2 J0

n :

0

f r z = z z 2h J0

a2r ! = n=1 fn z J0 anr ! :

1

X

f2 z = z z 2h

fn z 0 n 6=:2

2:5:50

1

1

d

r

d

J0 nr

Zn z + d2Zn J0

nr

+ fn z J0

nr

= 0 ,

8r

dr

!!

a

!

a

!9

n=1

dr

a

dz

X

2

=

:

8

9

1

nr

d2Zn

nr

nr

,

n=1

nJ0

a !

Zn z +

dz

2 J0

a !

+ fn z J0

a

!=

= 0 ,

X

:

, 1

8d2Z2n

n

2 Zn z

+ fn z 9 J0

nr

!

= 0:

n=1

dz

a !

=

a

X

:

n

6 , )

Z

z 7'

d2Zn

n

!

2

Zn z = fn z

n = 1 1:

2:5:51

dz2

a

2.5.48 2.5.46

Zn0 = 0

dZn h

= 0 n =

:

2:5:52

1 1

dz

!

Zn z 0. )

" ( # n = 2:

Z2

2

!

2 Z2 = z

z

2h 9

2:5:53

a

Z20 = Z2 h = 0:

=

2:5:54

,-. (. 2.5.53

- # !

Z0 z = A2 sh 2z

+ B2 ch

2 z h :

a

a

0 12 2.5.53 %

-

-

Z z = Az2 + Bz + C:

2:5:55

2.5.55 2.5.53

2A a2 !2

Az2 + Bz + C = z2 + 2hz:

, ( &

A =

a

!2 B = 2h

a

!2 C = 2

a

!4 :

2

2

2

4 ", -. 12 2.5.53 :

Z2 z = A2 sh 2z

+B2 ch 2 z h +

a

2 0z2 2hz + 2

a

21 : 2:5:56

2

!

2

a

a

@

!

A

2.5.56 2.5.54 , %

A2 B2:

A2 = 0

B2 =

2 a= 2 4

:

ch 2h=a

2 a= 2 4

2 z h

a

2

0z

2

2hz + 2

a

2

1 :

Z2 z =

ch

+

!

!

2:5:57

ch 2h=a

a

2

2

@

A

" 2.5.44 | 2.5.46 u r z,

2.5.48 , & Z2 z 2.5.57 ,

Zn z

0

n 6=.2

.

a

2

2

2a2

2 z h

2

a

23

2r

u r z =

!

ch

+ z

2hz + 2

!

J0

a ! :

2

2

2h

a

2

7

6

2

ch a !

4

5

2:5:58

2.5.2.

) *

4u = f r z & &

D = f r ' z : r a

. J0 i =

0 J1 i = 0:

1.

f r z = z z h J0

1r=a

u r=a= u z=0= u z=h= 0:

2.

f r z = z z 2h J0

1r=a

u

r=a

= u

z=0

= uz

z=h

= 0:

3.

f r z = z

3

3

J0

2r=a

u

= u

h

r=a

= uz

z=0

z=h

= 0:

2

4.

f r z = z

2z 3h J0 2r=a

u

r=a

= uz

z=0

= uz

z=h

= 0:

= 0:

5.

f r z = z h z z h=2J0 1r=a

ur

= u

= u

6.

f r z = z h

3

3

J0 2r=a

ur

r=a

z=0

z=h

+ h

r=a

= u

z=0

= uz

z=h

= 0:

2

2

7.

f r z = h

z

J0 3r=a

ur

r=a

= uz

z=0

= u

z=h

= 0:

8.

f r z = z

2

z h J0

2r=a

u

r=a

= u

z=0

= u

z=h

= 0:

9.

f r z = z 2h

z J0 2r=a u r=a= u z=0= uz z=h= 0:

10.

f r z = z2 z

3h=2 J0 1r=a

u

r=a

= uz

z=0

= uz

= 0:

z=h

11.

f r z = h3

z3 J0 1r=a

u

r=a

= uz

z=0

= u

z=h

= 0:

12.

f r z = h

z z2J0 2r=a

ur

r=a

= u

z=0

= u

z=h

= 0:

13.

f r z = z2 z

3h=2 J0 3r=a

ur

r=a

= u

z=0

= uz

z=h

= 0:

14.

f r z = z2

h2 J0 2r=a

ur

r=a

= uz

z=0

= u

z=h

= 0:

15.

f r z = z z

h 2J0 3r=a

u

r=a

= u

z=0

= u

z=h

= 0:

16.

f r z = z2 z

3h=2 J0 3r=a

u

r=a

= u

z=0

= uz

z=h

= 0:

17.

f r z = h2

z2 J0 3r=a

u

r=a

= uz

z=0

= u

z=h

= 0:

18.

f r z = z2

2z3=3h J0 3r=a u r=a= uz z=0= uz z=h= 0:

19.

f r z = z z

h 2J0 3r=a

ur

r=a

= u

z=0

= u

z=h

= 0:

20.

f r z = z 2h

z J0 3r=a ur r=a= u z=0= uz z=h= 0:

21.

f r z = z3

h3 J0 2r=a

ur

r=a

= uz

z=0

= u

z=h

= 0:

22.

f r z = h

z z2J0 1r=a

u

r=a

= u

z=0

= u

z=h

= 0:

23.

f r z = z

h 3 + h3 J0 2r=a

u

r=a

= u

z=0

= uz

= 0:

z=h

24.

f r z = z2

h2 J0 1r=a

u

r=a

= uz

z=0

= u

z=h

= 0:

25.

f r z = z2 z

h 2J0 2r=a

u

r=a

= uz

z=0

= uz

= 0:

z=h

26.

f r z = z z

h J0 1r=a ur r=a= u z=0= u z=h= 0:

27.

f r z = z z

2h J0 1r=a

u

r=a= u z=0= uz z=h= 0:

28.

f r z = h3

z3 J0 1r=a

ur

r=a

= uz

= u

z=h

= 0:

z=0

29.

f r z = z

h z z h=2J0 1r=a

u r=a= u z=0= u z=h= 0:

30.

f r z = 3h=2 z z2J0 1r=a

u

r=a

= u

z=0

= uz

z=h

= 0:

1

@

2 @u

1

@

@u

1

@2u

4u =

r

!+

sin

@ !+

= 0 2:6:1

r2

@r

@r

r2 sin

@

r2 sin2

@'2

' # ! #

u r=a= g = cos 2 + cos :

2:6:2

1 D = fr ag )! # !' !

ju r ' j 1

r ! 0*

2:6:3

2.12

u r ' = O 1r! r ! 1:

2:6:4

1

@

2 @u

1

@

@u

4u =

r

! +

sin

@ ! = 0:

2:6:5

r2

@r

@r

r2 sin

@

u r = R r Y :

2:6:6

2.6.6 2.6.5

:

R r

1

d

dY

!

r + 2

sin d

sin

d

r

= :

R r

=

Y

r2

$* *+ = 0 =

jY 0j 1 jY j 0:

2:6:11

0 & x = cos , ( 1 23 *

& Y = Y arccos x

~

Y x : & '

2.6.8 & ' & :

~

~

~

dY x =

dY x dx =

dY sin :

d

dx

d

dx

d

01

~

x

1

2

dY

~

x = 0

2:6:12

x

+ Y

dx @

dx

A

~

2:6:13

jY 1j 1:

2.6.12 , 2.6.13

n = n n + 1 n =

2:6:14

0 1

~

1

dn 2

n

1

dxn x 1 n =

Yn x = Pn x Pn x = 2n n!

0 :

,

2.6.8!, 2.6.11!

2:6:15!

Yn ! = Pncos ! n =

0 1

:

&

'( 2.6.7! = n:

2

n n + 1!

= 0 1:

Rn r! + rRn r!

r2

Rn r! = 0 n

Rn r! = Anrn + Bnr n+1:

2

/

r = 0 2.6.9! , Bn = 0 n =

0 1

. .

Rn r! = Anrn:

2

- / r !

1 2.6.10!

, An = 0 n = 0 1 .

2.6.5 , 2.6.2 , 2.6.3 .

,

! ",

" #

$$%

#

r :

1

u r = un r =

AnrnPncos :

2:6:16

1

g =

AnanPncos :

2:6:17

n=0

. * "

!

/ 0 # 1 1 12:

1

2

Z Pn x Pk x dx = Z Pncos Pkcos sin d = nk

:

2n

+ 1

1

0

Z g Pkcos sin d = Akak

2

k =

0 1

:

2k + 1

0

2k

+ 1

Ak =

Z g Pkcos sin d k = 0 1:

2:6:18

k

2a

0

cos + cos 2 = AnanPncos :

2:6:19

n=0

cos

-

2

1

3

2

1 + c

os

+ 2 cos = A

2C! + B cos

+

2C cos

A B C:

8 A

1

8 A =

1

2C = 1

3

B = 1

B = 1

3

4

2C = 2

C =

3:

$

:

:

%2.6.19)

1P0%cos

4P2%cos ) =

1

) + 1 P1%cos ) +

X

AnanPn%cos ):

3

3

n=0

* +

A0a0 = 1

A1a = 1

A2a2 = 4

An = 0

n 6=f0 1 2g: %2:6:20)

3

3

u%r ) =

1P0%cos

) + a 1rP1%cos ) +

4a 2r2P2%cos

) =

3

3

1

2

=

3

+ a

1r cos +

3a 2r2%3 cos2 1):

%2:6:21)

1

1

u%r ) =

X

un%r

) =

X

Bnr n+1 Pn%cos ):

%2:6:22)

n=0

n=0

2k + 1ak+1

Bk =

Z g Pkcos sin d :

2:6:23

2

0

! ! " # 2.6.5 , 2.6.2 , 2.6.4

' ()

cos + cos 2 =

Bna n+1 Pncos :

n=0

- #

1

4

1

P0

cos + 1 P1cos +

P2cos =

Bna

n+1

Pncos :

3

3

n=0

1-

B0a 1 =

1

B1a 2 = 1

B2a 3

= 4

Bn

= 0 n 6=f0 1 2g: 2:6:24

3

3

0 2.6.24 2.6.22 # ! ! " #

u r =

1

ar

1

2

r

2

P1cos +

4

3

r

3

P2

cos =

3

P0cos + a

3

a

=

1

1

2

2

2

3

3

2

3ar

+ a

r

cos

+

3a

r

3 cos

1:

2:6:25

. 1

! " # 2.6.21 :

1

1

2

2

2

2

u r =

3

+ a

r cos +

3a

r

3 cos

1 4

2 ! ! " # 2.6.25 :

1

1

2

2

2

3

3

2

u r =

3ar

+ a

r

cos

+

3a

r

3 cos

1: